Use of Uncertain Additional Information in Newsvendor Models

Abstract

The newsvendor problem is a popular inventory management problem in supply chain management and logistics. Solutions to the newsvendor problem determine optimal inventory levels. This model is typically fully determined by a purchase and sale prices and a distribution of random market demand. From a statistical point of view, this problem is often considered as a quantile estimation of a critical fractile which maximizes anticipated profit. The distribution of demand is a random variable and is often estimated on historic data. In an ideal situation, when the probability distribution of the demand is known, one can determine the quantile of a critical fractile minimizing a particular loss function. Since maximum likelihood estimation is asymptotically efficient, under certain regularity assumptions, the maximum likelihood estimators are used for the quantile estimation problem. Then, the Cramer-Rao lower bound determines the lowest possible asymptotic variance. Can one find a quantile estimate with a smaller variance then the Cramer-Rao lower bound? If a relevant additional information is available then the answer is yes. Additional information may be available in different forms. This manuscript considers minimum variance and minimum mean squared error estimation for incorporating additional information for estimating optimal inventory levels. By a more precise assessment of optimal inventory levels, we maximize expected profit.

Use of Uncertain External Information in Statistical Estimation

“…Overall, if δ=0 can be surely assumed, the minimum variance estimator θ 0 0 is to be used, and if some protection against possible bias (disinformation/misinformation) is needed then minimum MSE estimation with θ 0 δ is a better choice with the understanding that θ 0 δ is inferior to θ 0 0 under δ=0. The large sample distribution of θ 0 δ differs from normal, but is known, see (15). The estimator θ 0 δ can be used to evaluate the impact of bias on the estimating procedure.…”

“…This manuscript shows how external information on sales can be used under (1) the assumption that the external information came from an unbiased data source and (2) that the external data source can be very different to assume unbiasedness. This manuscript is an updated and extended version of a proceedings paper [15] where similar statistical methodology was applied to newsvendor-type problems. Section 3 presents main mathematical results for combining empirical and external data summarized by sample means and their variances.…”